CBSE · Class 11 · Mathematics · Relations and FunctionsThe function $f: \mathbb{N} \to \mathbb{N}$ defined by $f(x) = x^2$ is:
Step-by-Step Solution
For one-one: if $x_1^2 = x_2^2$, since $x_1, x_2 \in \mathbb{N}$, $x_1 = x_2$, so it is one-one. For onto: there is no natural number $x$ such that $f(x) = 2$ (since $\sqrt{2} \notin \mathbb{N}$), so it is not onto. Therefore, $f$ is one-one but not onto. Option B is correct.
Detailed Options Breakdown
Option : One-one and Onto
Incorrect choice. This distractor represents a common misunderstanding of the core principles of Relations and Functions.
Option 1: One-one but not Onto (Correct Answer)
Correct choice. Refer to the step-by-step verified solution guidelines above for details.
Option 2: Onto but not One-one
Incorrect choice. This distractor represents a common misunderstanding of the core principles of Relations and Functions.
Option 3: Neither One-one nor Onto
Incorrect choice. This distractor represents a common misunderstanding of the core principles of Relations and Functions.
💡 Study Guide: This question tests core syllabus concepts from Relations and Functions. For formulas, key summaries, and mock exam reference guides, read the full Relations and Functions Revision Notes.